Despite dramatic advances in the energy resolution and throughput of electron monochromators for high resolution electron energy loss spectroscopy (HREELS), a major limitation of conventional, dispersive sector, electron energy analyzers is that they are inherently serial devices, leading to long data acquisition times. The advantage of higher resolution leads to trade-offs in performance (throughput) because channel step size must be reduced, and therefore increasing the number of channels required to measure a given spectral region. Using a multi-channel plate detector to ameliorate this problem is one possibility. Indeed, time-resolved HREELS measurements have been demonstrated with a multi-channel plate in the dispersive plane of a conventional analyzer. However, parallel detection can be accomplished in this way only over a limited spectral range without degrading resolution. Thus, development of an analyzer based upon parallel detection would benefit both typical spectral investigations and allow new experiments to be performed, such as recent inelastic diffraction experiments which are both momentum and energy resolved.
Pseudo-random binary sequences (PRBS), also known as maximal length (ML) shift register sequences and/or pseudorandom noise (PN) sequences, have been used for modulation of photon and particle beams in widely used time-of-flight (TOF) techniques. These find application in, for example, neutron scattering, molecular beam scattering, and ion mass spectroscopy. The PRBS-TOF method achieves a throughput advantage over single pulse TOF due to the 50% duty cycle. PRBS modulation has been combined with TOF-MS, for example in a paper by Brown, W. L., et al., “Electronic sputtering of low temperature molecular solids,” in Nuclear Instruments and Methods in Physics Research, Vol. B1, 1984, pp. 307-314. In this example, an incident ion beam was pulsed with a pseudo-random sequence. The ion beam impinged upon a condensed water matrix sample, sputtering or producing secondary ions and neutrals which are measured in a time-of-flight detector. An electron impact ionizer was used to ionize the neutral and then a quadrupole filter was employed to mass select the products. Thus, in this case, the TOF technique was used to measure the energy distribution. To improve the signal-to-background ratio, PRBS modulation and cross-correlation recovery techniques were used, assuming that the source modulation was ideal.
More specifically, in such approaches, the underlying TOF spectrum (the object spectrum, o) is modulated with the PRBS sequence, p, resulting in a periodic, time sequence that is assumed to be defined mathematically as, (p{circle around (x)}o). In the standard cross-correlation recovery method, an estimate of the TOF spectrum, r, is obtained by correlating the detected TOF signal data with the PRBS modulation sequence, p: r=p⊕(p{circle around (x)}o). Here, {circle around (x)} and ⊕ denote convolution and correlation, respectively.
A special property of maximal length PRBS sequences is that the autocorrelation of the discrete binary sequence p is a substantially a delta function; therefore, the recovered spectrum, r, is substantially identical to the original object spectrum, o. In reality, the modulation function p is continuous, but r is an estimate of o as long as the time base (minimum pulse width) of the modulation function is small compared to the linewidth of the narrowest features in o. If this is not the case, then the throughput advantage is gained at the expense of resolution in the recovered spectrum, and over-sampling of the modulated signal, (p{circle around (x)}o), leads to a recovered spectrum which is the autocorrelation (p⊕p) (roughly, a triangular pulse) convoluted with the object function: r=(p⊕p){circle around (x)}o .
In fact, the modulation of the particle beam, whether performed at the source, with a spinning disk type of mechanical chopper, or with an electrostatic deflection based device, is at best described approximately as a convolution with the ideal sequence, (p{circle around (x)}o). First, the actual effect of the modulating device on the particle beam differs to some extent from the ideal sequence, p. A number of artifacts in the recovered object function, r, are well known in the art, and some types of non-ideal behavior can be corrected through post processing when (p⊕p) differs from a delta function, such as arises from machining errors in creating the slots in mechanic spinning disks. Second, most modulators do not act in exactly the same manner on different particles in the beam; for example, the finite thickness of spinning disks leads to a velocity dependent modulation function in molecular beam scattering applications. In this case, the assumption of a convolution is not strictly true.
To the extent that the modulation can be described by a convolution, and the actual modulation function, p, is known or can be estimated, the object function may be recovered simply by Fourier deconvolution. In practice, the presence of noise in measured data complicates deconvolution of spectral data in the simplest cases when the instrument function can be described by a single feature.
The deconvolution of a PRBS modulation sequence, in which the data contains multiple overlapping copies of the underlying object function, has not been reported in spectroscopic applications, to our knowledge.
Probability-based estimation methods for recovery of one-dimensional distributions, and for resolution enhancement of one-dimensional spectral data and two-dimensional image data, have been used by astronomers since 1972. (See Richardson, W. H. 1972, “Bayesian-Based Iterative Method of Image Restoration”, J. Opt. Soc. Am. 62, 55-59; Frieden, B. R. 1972, “Restoring with Maximum Entropy and Maximum Likelihood”, J. Opt. Soc. Am. 62, 511-18; Lucy, L. B. 1974, “An iterative technique for the rectification of observed distributions”, Astron. J. 19, 745-754; and Ables, J. G. 1974, “Maximum Entropy Spectral Analysis”, Astron. Ap. Suppl. 15, 383-93.) Recent success with iterative maximum likelihood and Bayesian methods has been demonstrated in a paper by Frederick, B. G., et al., entitled “Spectral restoration in HREELS,” in the Journal of Electron Spectroscopy and Related Phenomena, Vol. 64/65, 1993, pp. 825. The maximum likelihood result is simply an array, which convoluted with the modulation function, fits the data as well as possible, given the noise distribution. A well known example of this approach is the algorithm reference in the paper by L. B. Lucy. The Bayesian method employed by Frederick, et al., includes a maximum entropy constraint that limits the degree of resolution enhancement in a manner that leads to a single converged estimate with no arbitrary adjustable parameters.